Relation between moduli spaces and classifying spaces

Question

I hope this question is suitable to be posted here on MO.

I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider myself as someone with an expert level knowledge in any of these objects, but let me explain what I think.

The classifying space of a group/H-group, or a monoial category, most of the time is ought to be a representing object for a cohomology theory. The word classifying here, is mainly referring to the fact that $BG$ does parametrise certain ``bundles'' over paracompact spaces with a $G$-structure. The moduli space, is also ought to be a space which somehow parametrises the class of objects in a certain category - hope this analogy is not too vague. It somehow puts the totality of certain object in a `topological space'. Hence, both of these spaces are meant to parametrise certain objects.

Of course, I have not declared which definitions I take for these object, and I hope there is a common understanding about these spaces when an example is given. My main question is this. If $C$ is a category for which I can associated a classifying space $BC$ as well as a moduli space to the class of its objects, say $\mathcal{M}_C$. Then, is there a map

$$BC\to\mathcal{M}_C$$

or the way around?

Let me add that the work of Madsen and Weiss, on the stable Mumford conjecture, provides an example of the relation that I have asked above. They show that the infinite loop space associated to a certain spectrum $\mathbb{C} P_{-1}$ is related to the classifying space of the cobordism category of oriented surfaces, say $BC_2$ (in fact there is a weak homotopy equivalence) which is meant to solve Mumford's conjecture as $\Omega^{\infty-1}\mathbb{C} P_{-1}$ and the moduli space of Riemannian surfaces are ought to be the same?! But, I still don't know where a map such as $\Omega^\infty\mathbb{C}P_{-1}\to \mathcal{M}_\infty$ or a map $BC_2\to \mathcal{M}_\infty$ is constructed where $\mathcal{M}_\infty$ is the moduli space of Riemannian surfaces of arbitrary genus. I am aware of the map Madsen-Tillmann map $\mathbb{Z}\times B\Gamma_\infty^+\to\Omega^\infty\mathbb{C}P_{-1}$ with $\Gamma_\infty$ is the stable mapping class group and $(-)^+$ is Qullien's plus construction. The work of Galatius-Madsen-Tillmann-Weiss also provides $\alpha_\infty:BC_2\to\Omega^{\infty-1}\mathbb{C}P_{-1}$. But, I don't think of the source nor the target as a moduli space.

I will be very grateful for any advise. Please let me know if I am mixing some objects.

Thanks.

Answer 1

There is a sense in which the relation between moduli stacks and classifying spaces can be formalized, at least when we use smooth manifolds as parametrizing objects. (Topological manifolds and PL-manifolds also suffice.)

Start with a stack F of spaces on the site of smooth manifolds, which we think of as the moduli stack of some smoothly parametrized objects.

Define the concordification CF of F as the prestack CF(X) := hocolim_{n∈Δ^op} F(Δ^n×X), where Δ^n is the smooth extended n-simplex, i.e., the smooth manifold R^n, with the appropriate face and degeneracy maps.

(The name concordification comes from the fact that F[X] := π_0(CF(X)) is the set of concordance classes of sections of F over X, where two sections x and y over X are concordant if there is a section z over R×X whose restrictions to 0×X and 1×X are isomorphic to x and y respectively.)

It is a nontrivial result that for any stack F of spaces the prestack CF is actually a stack. (For the much easier case of a stack (sheaf) of sets this is shown in Proposition 2.17 and Proposition A.1 in the cited paper of Madsen and Weiss.)

Furthermore, CF is concordance-invariant: for any manifold X the canonical pullback map F(X)→F(R×X) is a weak equivalence.

For any stack (or prestack) of spaces G one can construct a natural map G(X)→Map(X,CG(pt)). Indeed, consider the functor C(F) := (CF)(pt) from prestacks to spaces. This functor is enriched over spaces, so we have a map Map(X,G)→Map(CX,CG). Here the left Map denotes the mapping space of stacks (which are enriched over spaces) and X denotes the representable stack of X. The right Map is simply a mapping space of spaces. By the (enriched) Yoneda lemma we have Map(X,G)=G(X). Furthermore, CX is simply the (smooth) singular simplicial set of X, i.e., the underlying homotopy type of X. Abusing the notation, we write Map(CX,CG)=Map(X,CG)=Map(X,CG(pt)). Altogether, we have a map G(X)→Map(X,CG(pt)).

An observation that goes back at least to Morel and Voevodsky (see Proposition 3.3.3 in their paper) says that concordance-invariant stacks of spaces on the site of smooth manifolds are precisely locally constant stacks, i.e., the canonical map G(X)→Map(X,G(pt)) is a natural weak equivalence for any concordance-invariant stack of spaces G.

In particular, in our case we get a natural weak equivalence CF(X)→Map(X,CF(pt)). Taking π_0 on both sides we get an isomorphism of sets F[X]→[X,CF(pt)].

In other words, for any (moduli) stack of spaces F the set F[X] of concordance classes of sections of F over X is naturally isomorphic to [X,CF(pt)], where CF(pt) = hocolim_{n∈Δ^op} F(Δ^n). It is natural to call CF(pt) the classifying space of F.

For example, for F(X)=Ω^n_cl(X), the set of closed n-forms on X, we get CF(pt)=K(R,n), the nth Eilenberg—MacLane space of R (this is covered by the result of Madsen and Weiss because F(X) is a set).

If we take F(X) to be (the nerve of) the groupoid of real vector bundles of dimension n over X equipped with a connection (and isomorphisms that preserve connections), then CF(pt)=BO(n), the classifying space of n-dimensional vector bundles (or concordance classes of bundles with connection; two bundles with connection are concordant if and only if they are isomorphic as bundles without connection). Note that we get the same answer if we take vector bundles without connection; the concordification construction does not see any local data such as connections, forms, etc.

If F(X) is the space of bundle (n−1)-gerbes with (or without) connection, then CF(pt)=B^n U(1), the classifying space of bundle (n−1)-gerbes (or concordance classes of bundle (n−1)-gerbes with connection).

One can also apply the concordification functor C to morphisms of stacks, for example, applying C to the Chern—Weil construction recovers Chern classes, etc.